Automorphisms of Generalized Down-Up Algebras

A generalization of down-up algebras was introduced by Cassidy and Shelton (2004 Cassidy , T. , Shelton , B. ( 2004 ). Basic properties of generalized down-up algebras . J. Algebra 279 ( 1 ): 402 – 421 .[Crossref], [Web of Science ®] , [Google Scholar]), the so-called “generalized down-up algebras”. We describe the automorphism group of conformal Noetherian generalized down-up algebras L(f, r, s, γ) such that r is not a root of unity, listing explicitly the elements of the group. In the last section, we apply these results to Noetherian down-up algebras, thus obtaining a characterization of the automorphism group of Noetherian down-up algebras A(α, β, γ) for which the roots of the polynomial X ² ? α X ? β are not both roots of unity.     

Equivalences for Weak Crossed Products

In this article, we give a criterion that characterizes equivalent weak crossed products. By duality, we obtain a similar result for weak crossed coproducts and, as a consequence, we find the conditions that assures the equivalence between two weak crossed biproducts. As an application, we show that the main results proved by Panaite in [12 Panaite, F. (2014). Equivalent crossed products and cross product bialgebras. Commun. Algebr. 42:1937–1952.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] (see also [11 Panaite, F. (2012). Invariance under twisting for crossed products. Proc. Am. Math. Soc. 140:755–763.[Crossref], [Web of Science ®] , [Google Scholar]]), for Brzeziński's crossed products, admits a substantial reduction in the imposed conditions.     

Remarks on the Global Solutions of 3-D Navier-Stokes System with One Slow Variable

By applying Wiegner's method in [16 Wiegner , M. ( 1987 ). Decay results for weak solutions to the Navier-Stokes equations on ? ⁿ . J. London Math. Soc. 35 : 303 – 313 .[Crossref], [Web of Science ®] , [Google Scholar]], we first prove the large time decay estimate for the global solutions of a 2.5 dimensional Navier-Stokes system, which is a sort of singular perturbed 2-D Navier-Stokes system in three space dimension. As an application of this decay estimate, we give a simplified proof for the global wellposedness result in [6 Chemin , J.-Y. , Gallagher , I. ( 2010 ). Large, global solutions to the Navier-Stokes equations, slowly varying in one direction . Transactions of the American Mathematical Society 362 : 2859 – 2873 .[Crossref], [Web of Science ®] , [Google Scholar]] for 3-D Navier-Stokes system with one slow variable. Let us also mention that compared with the assumptions for the initial data in [6 Chemin , J.-Y. , Gallagher , I. ( 2010 ). Large, global solutions to the Navier-Stokes equations, slowly varying in one direction . Transactions of the American Mathematical Society 362 : 2859 – 2873 .[Crossref], [Web of Science ®] , [Google Scholar]], here the assumptions in Theorem 1.3 are weaker.     

Derived Representation Type of Schur Superalgebras

In this note we extend the results of Bekkert and Futorny in [2 Bekkert , V. , Futorny , V. ( 2003 ). Derived categories of Schur algebras . Comm. Alg. 31 : 1799 – 1822 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] and Hemmer, Kujawa and Nakano in [10 Hemmer , D. J. , Kujawa , J. , Nakano , D. K. ( 2006 ). Representation types of Schur superalgebras . J. Group Theory 9 : 283 – 306 .[Crossref], [Web of Science ®] , [Google Scholar]] and determine the derived representation type of Schur superalgebras.     

Uniqueness Criteria for the Adjoint Equation in State-Constrained Elliptic Optimal Control

The article considers linear elliptic equations with regular Borel measures as inhomogeneity. Such equations frequently appear in state-constrained optimal control problems. By a counter example of Serrin [18 J. Serrin ( 1964 ). Pathological solutions of elliptic differential equations . Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 : 385 – 388 . [Google Scholar]], it is known that, in the presence of non-smooth data, a standard weak formulation does not ensure uniqueness for such equations. Therefore several notions of solution have been developed that guarantee uniqueness. In this note, we compare different definitions of solutions, namely the ones of Stampacchia [19 G. Stampacchia ( 1965 ). Le probléme de Dirichlet pour les équations elliptiques du second ordre à coéffcients discontinus . Ann. Inst. Fourier 15 : 189 – 258 .[Crossref] , [Google Scholar]] and Boccardo-Galouët [4 L. Boccardo and T. Gallouët ( 1989 ). Nonlinear elliptic and parabolic equations involving measure data . J. Func. Anal. 87 : 149 – 169 .[Crossref], [Web of Science ®] , [Google Scholar]] and the two notions of solutions of [2 J.-J. Alibert and J.-P. Raymond ( 1997 ). Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls . Numer. Func. Anal. Optim. 18 : 235 – 250 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 7 E. Casas (1993). Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31:993–1006.[Crossref], [Web of Science ®] , [Google Scholar]], and show that they are equivalent. As side results, we reformulate the solution in the sense of [19 G. Stampacchia ( 1965 ). Le probléme de Dirichlet pour les équations elliptiques du second ordre à coéffcients discontinus . Ann. Inst. Fourier 15 : 189 – 258 .[Crossref] , [Google Scholar]], and prove the existence of solutions in the sense of [2 J.-J. Alibert and J.-P. Raymond ( 1997 ). Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls . Numer. Func. Anal. Optim. 18 : 235 – 250 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 4 L. Boccardo and T. Gallouët ( 1989 ). Nonlinear elliptic and parabolic equations involving measure data . J. Func. Anal. 87 : 149 – 169 .[Crossref], [Web of Science ®] , [Google Scholar], 7 E. Casas (1993). Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31:993–1006.[Crossref], [Web of Science ®] , [Google Scholar]] in case of mixed boundary conditions.     

Regularity of Analytic Algebra of Prime Characteristic

Let A be an analytic algebra over a field k of characteristic p > 0. In this article, for an analytic k-algebra we introduce the concept of analytic pⁿ-basis which generalizes the pⁿ-basis defined in [1 Furuya , M. , Niitsuma , H. ( 2002 ). Regularity criterion of Noetherian local rings of prime characteristic . J. Algebra 247 : 219 – 230 .[Crossref], [Web of Science ®] , [Google Scholar]], and the concept of an pⁿ-admissible field for an algebraic function field, we give regularity criteria and absolute regularity criteria for an analytic algebra A/k in terms of the higher differential algebra and analytic pⁿ-basis. The results are partial extension of our previous results for affine algebras to the case of analytic algebras (cf. [1 Furuya , M. , Niitsuma , H. ( 2002 ). Regularity criterion of Noetherian local rings of prime characteristic . J. Algebra 247 : 219 – 230 .[Crossref], [Web of Science ®] , [Google Scholar], 3 Furuya , M. , Niitsuma , H. (2006). Regular local rings essentially of finite type over fields of prime characteristic. J. Algebra 306:703–711.[Crossref], [Web of Science ®] , [Google Scholar]]), and these are partial generalization of results of Orbanz in the analytic case (cf. [9 Orbanz , U. ( 1973 ). Höhere Derivationen und Regularität . J. Reine Angew. Mathematik 262/263 : 194 – 204 .[Crossref] , [Google Scholar]]).     

The Complement Graph for Gaussian Integers Modulo n

This article is a continuation for the work done in [1 Abu Osba , E. , Al-Addasi , S. , Abu Jaradeh , N. ( 2008 ). Zero divisor graph for the ring of Gaussian integers modulo . n. Comm. Algebra 36 ( 10 ): 3865 – 3877 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 2 Abu Osba , E. , Al-Addasi , S. , Al-Khamaiseh , B. ( 2011 ). Some properties of the zero divisor graph for the ring of gaussian integers modulo . n. Glasgow Journal of Mathematics 53 : 391 – 399 .[Crossref], [Web of Science ®] , [Google Scholar]] on the zero divisor graph for the ring of Gaussian integers modulo n. It investigates when the complement graph of the zero divisor graph for the Gaussian integers modulo n connected, planar, regular, or Eulerian. The girth and diameter were also studied.     

Root extraction in one-relator groups and slenderness

In this note we strengthen a result of Newman and use it to prove a conjecture of Nakamura stated in [10 Nakamura, J. (2015). Atomic properties of the Hawaiian earring group for HNN extensions. Commun. Algebra 43:4138–4147.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] that torsion-free one-relator groups are noncommutatively slender.     

Certain Cyclically Presented Groups with the Same Polynomial

We consider three infinite families of cyclic presentations of groups, depending on a finite set of integers and having the same polynomial. Then we prove that the corresponding groups with the same parameters are isomorphic, and that the groups are almost all infinite. Finally, we completely compute the maximal Abelian quotients of such groups, and show that their HNN extensions are high-dimensional knot groups. Our results contain as particular cases the main theorems obtained in two nice articles: Johnson et al. (1999 Johnson , D. L. , Kim , A. C. , O'Brien , E. A. ( 1999 ). Certain cyclically presented groups are isomorphic . Comm. Algebra 27 ( 7 ): 3531 – 3536 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Havas et al. (2001 Havas , G. , Holt , D. F. , Newman , M. F. ( 2001 ). Certain cyclically presented groups are infinite . Comm. Algebra 29 ( 11 ): 5175 – 5178 . [CSA] [CROSSREF] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).     

On the Associated Primes of Matlis Duals of Local Cohomology Modules II

In continuation of [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] we study associated primes of Matlis duals of local cohomology modules (MDLCM). We combine ideas from Helmut Zöschinger on coassociated primes of arbitrary modules with results from [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] 4-6 Hellus , M. , Stückrad , J. ( 2008 ). On endomorphism rings of local cohomology modules . Proceedings of the American Mathematical Society 136 : 2333 – 2341 . Hellus , M. , Stückrad , J. ( 2008 ). Matlis duals of top local cohomology modules . Proceedings of the American Mathematical Society 136 : 489 – 498 . Hellus , M. , Stückrad , J. ( 2009 ). Artinianness of local cohomology . Journal of Commutative Algebra 1 : 269 – 274 . ], and obtain partial answers to questions which were left open in [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. These partial answers give further support for conjecture (*) from [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] on the set of associated primes of MDLCMs. In addition, and also inspired by ideas from Zöschinger, we prove some non-finiteness results of local cohomology.     

A new family of set-theoretic solutions of the Yang–Baxter equation

A new family of non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation is constructed. Two subfamilies, consisting of irretractable square-free solutions, are new counterexamples to Gateva-Ivanova’s Strong Conjecture [7 Gateva-Ivanova, T. (2004). A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation. J. Math. Phys. 45(10):3828–3858.[Crossref], [Web of Science ®] , [Google Scholar]]. They are in addition to those obtained by Vendramin [15 Vendramin, L. (2016). Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova. J. Pure Appl. Algebra 220:2064–2076.[Crossref], [Web of Science ®] , [Google Scholar]] and [1 Bachiller, D., Cedó, F., Jespers, E., Okniński, J. (2017). A family of irretractable square-free solutions of the Yang-Baxter equation. Forum Math. (to appear). [Google Scholar]].     

Strongly J-Clean Matrices Over Local Rings

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly J-clean 2 × 2 matrix over a noncommutative local ring. The criteria on strong J-cleanness of 2 × 2 matrices in terms of a quadratic equation are given. These extend the corresponding results in [8 Li , B. ( 2009 ). Strongly clean matrix rings over noncommutative local rings . Bull. Korean Math. Soc. 46 : 71 – 78 .[Crossref], [Web of Science ®] , [Google Scholar], Theorems 2.7 and 3.2], [9 Li , Y. ( 2007 ). Strongly clean matrix rings over local rings . J. Algebra 312 : 397 – 404 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 2.6], and [11 Yang , X. , Zhou , Y. ( 2008 ). Strongly cleanness of the 2 × 2 matrix ring over a general local ring . J. Algebra 320 : 2280 – 2290 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 7].     

Classification of Pointed Fusion Categories of dimension 8 up to weak Morita equivalence

In this paper we give a complete classification of pointed fusion categories over ? of global dimension 8. We first classify the equivalence classes of pointed fusion categories of dimension 8, and then we proceed to determine which of these equivalence classes have equivalent categories of modules following the procedure presented in [9 Naidu, D. (2007). Categorical Morita equivalence for group-theoretical categories. Commun. Algebra 35(11):3544–3565.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 11 Uribe, B. (2017). On the classification of pointed fusion categories up to weak Morita equivalence. Pac. J. Math. 290(2):437–466.[Crossref], [Web of Science ®] , [Google Scholar]]. The results of this paper permit to recover the classification of twisted quantum doubles of groups of order 8 up to gauge equivalence of braided quasi-Hopf algebras that was previously done in [6 Mason, C., Ng, S.-H (2001). Group cohomology and gauge equivalence of some twisted quantum doubles. Trans. Am. Math. Soc. 353(9):3465–3509.[Crossref], [Web of Science ®] , [Google Scholar]] and [5 Goff, C., Mason, G., Ng, S.-H (2007). On the gauge equivalence of twisted quantum doubles of elementary abelian and extra-special 2-groups. J. Algebra 312(2):849–875.[Crossref], [Web of Science ®] , [Google Scholar]].     

Corrigendum To: Goldie Extending Modules

This corrigendum is written to correct the proof of Theorem 5.3 of Akalan et al. [1 Akalan , E. , Birkenmeier , G. F. , Tercan , A. ( 2009 ). Goldie extending modules . Comm. Algebra 37 : 663 – 683 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

Remarks on Reflexive Subcategories

In this article, we study reflexive subcategories of the category of finitely generated modules over Cohen–Macaulay local rings R. One of the main results yields a complete classification of the reflexive subcategories for the case when R is a one-dimensional complete local hypersurface of finite Cohen–Macaulay representation type. The other result yields a one-to-one correspondence between the subclasses of a stable category and the reflexive subcategories over a complete local hypersurface. Then, we attempt to extend the result in Takahashi [8 Takahashi, R. (2006). Remarks on modules approximated by G-projective modules. J. Algebra 301:748–780.[Crossref], [Web of Science ®] , [Google Scholar], Proposition 4.4] to a reflexive subcategory.     

Relationship between Rings and Hyperrings by Using the Notion of Fundamental Relations

In the theory of hyperrings, fundamental relations make a connection between hyperrings and ordinary rings. The fundamental relation α on hyperrings is studied by Davvaz and Vougiouklis in Communications in Algebra [8 Davvaz , B. , Vougiouklis , T. ( 2007 ). Commutative rings obtained from hyperrings (H _v -rings) with α*-relations . Comm. Algebra 35 ( 11 ): 3307 – 3320 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. Now, in this article, we obtain more results about this relation. In particular, we determine necessary and sufficient conditions such that the relation α is transitive.     

On Prime Spectrums of Commutative Rings

We show that π-regular rings and clean rings can be completely characterized by topological properties of their prime spectrums respectively. In addition, we give some applications of those result. Among others, we improve the main result of Samei (2004 Samei , K. ( 2004 ). Clean elements in commutative reduced rings . Comm. Algebra 32 ( 9 ): 3479 – 3486 . [CSA] [CROSSREF] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and give a new criterion for a clean ring that a commutative ring is clean if and only if idempotents lifts modulo every radical ideal.     

Derivations of a Restricted Weyl-Type Algebra Containing the Polynomial Ring

Buchsbaum and Eisenbud [3 Buchsbaum , D. , Eisenbud , D. ( 1977 ). Algebra structures for finite free resolutions and some structure theorems for ideals of codimension 3 . Amer. J. Math. 99 : 447 – 485 .[Crossref], [Web of Science ®] , [Google Scholar]] proved a structure theorem for Gorenstein ideals of grade 3 which claims that every Gorenstein ideal of grade 3 is generated by the submaximal order pfaffians of an alternating matrix. In this article, we characterize some classes of grade 3 perfect ideals which are linked to an almost complete intersection of even type. Furthermore, these structure theorems give us relations between the Gorenstein sequences and the Hilbert functions of the grade 3 perfect ideals that we characterized.     

Central Extensions and Derivations of the Lie Algebras of Skew Derivations for the Quantum Torus

ABSTRACT

In this article, we study the central extensions and derivations of the Lie algebra of skew derivations for the quantum torus. The results of the article generalize those obtained in Jiang and Meng (1998a Jiang , C. , Meng , D. ( 1998a ). The derivation algebra of the associative algebra C _q [X, Y, X ^?1, Y ^?1] . Comm. Algebra 26 : 1723 – 1736 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] b Jiang , C. , Meng , D. ( 1998b ). The automorphism group of the derivation algebra of the Virasoro-like algebra . Adv. Math. (China) 27 : 175 – 183 . [CSA]  [Google Scholar]) and Kirkman et al. (1994 Kirkman , E. , Procesi , C. , Small , L. ( 1994 ). A q-analog for the Virasoro algebra . Comm. Algebra 22 : 3755 – 3774 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).     

Supercharacter Theory Constructions Corresponding to Schur Ring Products

Diaconis and Isaacs have defined the supercharacter theories of a finite group to be certain approximations to the ordinary character theory of the group [7 Diaconis , P. , Isaacs , I. M. ( 2008 ). Supercharacters and superclasses for algebra groups . Trans. Amer. Math. Soc. 360 : 2359 – 2392 .[Crossref], [Web of Science ®] , [Google Scholar]]. We make explicit the connection between supercharacter theories and Schur rings, and we provide supercharacter theory constructions which correspond to Schur ring products of Leung and Man [12 Leung , K. H. , Man , S. H. ( 1996 ). On Schur rings over cyclic groups, II . J. Algebra 183 : 273 – 285 .[Crossref], [Web of Science ®] , [Google Scholar]], Hirasaka and Muzychuk [10 Hirasaka , M. , Muzychuk , M. ( 2001 ). An elementary abelian group of rank 4 is a CI-group . J. Combin. Theory Ser. A 94 : 339 – 362 .[Crossref], [Web of Science ®] , [Google Scholar]], and Tamaschke [20 Tamaschke , O. ( 1970 ). On Schur-rings which define a proper character theory on finite groups . Math. Z. 117 : 340 – 360 .[Crossref], [Web of Science ®] , [Google Scholar]].